# 10. Stellar Metallicity Distribution Functions¶

VICE’s singlezone and multizone objects automatically determine normalized stellar metallicity distribution functions (MDFs) for each simulation. The MDF, in its most general form, is given by:

$\frac{dN}{dZ} = \frac{\dot{N}}{\dot{Z}} \propto \frac{\dot{M}_\star}{\dot{Z}}$

This is fairly intuitive; the number of stars that form at a metallicity $$\approx$$ Z is proportional to the star formation rate at that time and inversely related to the rate at which the metallicity is evolving away from that value. VICE converts MDFs to probability distribution functions by ensuring that the integral over the bins is equal to one:

$\frac{dN}{d[X/Y]} \rightarrow \frac{ dN/d[X/Y] }{ \int dN } = \frac{ dN/d[X/Y] }{ \int_{-\infty}^{\infty} \frac{dN}{d[X/Y]} d[X/Y] }$

Note

In its current version, VICE only reports MDFs at the final timestep of the simulation.

In practice, the user specifies an array of bin-edges that they would like the MDF sorted into, and VICE creates arrays of zeroes whose lengths are the number of bins in the user’s array. In a singlezone simulation, the appropriate bins for each combination of [X/H] and [X/Y] are incremented by the star formation rate. At the final timestep, the normalization of the i’th bin is then approximated numerically by:

$\frac{\Delta N_i}{\Delta [X/Y]_i} \rightarrow \frac{ \Delta N_i / \Delta [X/Y]_i }{ \sum_j \frac{\Delta N_j}{\Delta [X/Y]_j} \Delta [X/Y]_j } = \frac{ \Delta N_i / \Delta [X/Y]_i }{ \sum_j \Delta N_j }$

The fraction of stars in a given range $$\Delta [X/Y]$$ is then given by the value of the reported MDF times $$\Delta [X/Y]$$.

In a multizone simulation, the metallicity distribution function is calculated directly from the star particles that are in a given zone at a given time. For each star particle in a given zone, the appropriate bins in [X/H] and [X/Y] are incremented by the mass of the star rather than by the star formation rate at previous timesteps. The same normalization process is then applied.